Algebra. 315. SQUARE Root AND AREA. 1. If a piece of land containing 768 square rods is three times as long as it is wide, how wide is it ? * Let x = the width, then 3.x = the length, 3x = 768 x = 16 How many 2. If a certain room is twice as long as it is wide, and the area of the floor 968 square feet, what is the length and the breadth of the room? 3. One half of the length of Mr. Smith's farm is equal to its breadth. The farm contains 80 acres. rods of fence will be required to enclose it ? 4. Each of four of the faces of a square prism is an oblong whose length is twice its breadth. The area of one of these oblongs is 72 square inches. What is the solid content of the prism ? † 5. The width of a certain field is to its length as 2 to 3. Its area is 600 square rods. The perimeter of the field is how many rods? 6. If f of the length of an oblong equals the width and its area is 768 square inches, what is the length of the oblong? 7. If to 21 times the square of a number you add 15 the sum is 375. What is the number? * To solve this problem arithmetically, one must discern that this piece of land can be divided into three equal squares, the side of each square being equal to the width of the piece. + Let no pupil attempt to solve this problem without first bringing into consciousness an image of the prism. Algebra. 316. SQUARE ROOT AND PROPORTION. When the same number forms the second and the third term of a proportion it is called a mean proportional, of the first and the fourth term; thus, in the proportion 3:6::6:12, 6 is a mean proportional of 3 and 12. EXAMPLE In the proportion 12:x::x:75, find the value of a Since the product of the means equals the product of the extremes, x times x equals 12 times 75, 01 x2 = 900 36 Find the value of x in each of the following proportions: 1. 9:x::x:16. 4. 12:x::x: 48. 3. 8:r::r: 32. 6. 36 :r::r: 49. 7. An estate was to be divided so that the ratio of A's part to B's would equal the ratio of B's part to C's. If A received $8000 and C received $18000, how much should B receive ? 8. Find the mean proportional off and 1%. 9. The ratio of the areas of two squares is as 4 to 9. What is the ratio of their lengths ? 10. The ratio of the lengths of two squares is as 9 to 16. What is the ratio of their areas? 11. The area of the face of one cube is to the area of the face of another cube as 16 to 25. What is the ratio of the solid contents of the cubes ? A B) Geometry. 317. RIGHT TRIANGLES. Fig. 1. 1. The longest side of a right-triangle is the hypothenuse. Either of the other sides may be regarded as the base and the remaining side as the perpendicular. In Fig. 1, AC is the ; BC, the and AB the 2. Convince yourself by examination Fig. 2. of the figures here given and by careful 2 measurements and paper cutting, that the square of the hypothenuse of a right H triangle is equivalent to the sum of the squares of the other two sides. 4 Figures 2 and 3 are equal squares. If from 3 figure 2, the four right-triangles, 1, 2, 3, 4, be taken, H, the square of the hypothenuse, remains. Fig. 3. If from figure 3, the four right-triangles (equal to 2 P the four right-triangles in figure 2) be taken, B, the square of the base, and P, the square of the 3 perpendicular, remain. When equals are taken B from equals the remainders are equal, therefore the square, H, equals the sum of the squares B 4 and P. 3. To find the hypothenuse of a right-triangle when the base and perpendicular are given: Square the base; square the perpendicular; extract the square root of the sum of these squares. 4. Give a rule for finding the perpendicular when the base and hypothenuse are given. 318. MISCELLANEOUS REVIEW. 1. Find approximately the diagonal of a square whose side is 20 feet. * 2. Find approximately the distance diagonally across a rectangular floor, the length of the floor being 30 feet and its breadth 20 feet. 3. How long a ladder is required to reach to a window 25 feet high if the foot of the ladder is 6 feet from the building and the ground about the building level? 4. If the length of a rectangle is a, and its breadth, b, what is the diagonal ? 5. The base of a right-triangle is 40 rods and its perpendicular, 60 rods. (a) What is its hypothenuse? (b) What is its area ? (c) What is its perimeter ? 6. The area of a certain square piece of land is 23 acres. (a) Find (in rods) its side. (b) Find its perimeter. (c) Find its diagonal, true to tenths of a rod. 7. The length of a rectangular piece of land is to its breadth as 4 to 3. Its area is 30 acres. (a) Find its breadth. (b) Find its perimeter. (c) Find the distance diagonally across it. 8. A certain piece of land is in the shape of a righttriangle. Its base is to its altitude as 3 to 4. Its area is 96 square rods. (a) Find the base. (b) Find the altitude. (c) Find the perimeter. 9. Find one of the two equal factors of 93025. 10. Find one of the three equal factors of 74088. (P. 366.) * From the study of right-triangles on page 219 it may be learned that the diagonal of a square is equal to the square root of twice the square of its side. METRIC SYSTEM. NOTE. - Work equivalent to that found on pp. 154, 164, 174, etc., of the Werner Arithmetic, Book II., should be done by the pupils before this chapter is attempted. If such work has not been done, the teacher should present the subject orally in the order given on the pages named. Pupils must learn to think of quantity in metric units. 319. All units in the metric system of measures and weights are derived from the primary unit known as the meter. When the length of the primary unit of this system was determined it was supposed to be one ten-millionth of the distance from the equator to the pole. A pendulum that vibrates seconds is nearly one meter long. * In the names of the derived units of this system the prefix deka means 10; hekto means 100; kilo means 1000; myria means 10000; deci means tenth ; centi means hundredth ; milli means thousandth. * The teacher must see that a meter stick is provided and that the pupils use it in measurement until they can think its multiples and its divisors without reference to other units of measurement. + In the common pronunciation of these words the primary accent is on the first syllable and a secondary accent on the penultimate syllable; thus, cen'timeter. In the better pronunciation the accent is on the vowel preceding the letter m, that is, on the antepenultimate syllable: thus, centim'eter, dekam'eter, etc. |